Abstract
There is no unique way to quantify the degree of delocalization of quantum states in unbounded continuous spaces. In this work, we explore a recently introduced localization measure that quantifies the portion of the classical phase space occupied by a quantum state. The measure is based on the α-moments of the Husimi function and is known as the Rényi occupation of order α. With this quantity and random pure states, we find a general expression to identify states that are maximally delocalized in phase space. Using this expression and the Dicke model, which is an interacting spin-boson model with an unbounded four-dimensional phase space, we show that the Rényi occupations with α>1 are highly effective at revealing quantum scars. Furthermore, by analyzing the high moments (α>1) of the Husimi function, we are able to identify qualitatively and quantitatively the unstable periodic orbits that scar some of the eigenstates of the model.
Highlights
In the classical domain, typical trajectories of chaotic systems fill the available phase space
Generalized inverse participation ratios and Renyi entropies of order α quantify the degree of delocalization of quantum states in Hilbert space
We employed a measure of localization called Renyi occupation of order α, which is defined over classical energy shells and is based on the α-moments of the Husimi function
Summary
Typical trajectories of chaotic systems fill the available phase space. We obtain a general analytical expression for the Renyi occupations of order α using maximally delocalized random pure states We employ this expression to analyze the eigenstates of the Dicke model in the chaotic regime and distinguish the eigenstates that are maximally delocalized from those that are highly localized in phase space. The analysis of high-moments (α > 1) of the Husimi functions of these highly localized states allow us to identify more clearly the classical unstable periodic orbits that cause their scarring. These orbits are different from the ones found in Ref.
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